Geochemistry, Geophysics, Geosystems

Factors controlling the mode of rift interaction in brittle-ductile coupled systems: A 3D numerical study

Authors


Abstract

[1] The way individual faults and rift segments link up is a fundamental aspect of lithosphere extension and continental break-up. Little is known however about the factors that control the selection of the different modes of rift interaction observed in nature. Here we use state-of-the-art large deformation 3D numerical models to examine the controls on the style and geometry of rift linkage between rift segments during extension of crustal brittle-ductile coupled systems. We focus on the effect of viscosity of the lower layer, the offset between the rift basins and the amount of strain weakening on the efficiency of rift linkage and rift propagation and the style of extension. The models predict three main modes of rift interaction: 1) oblique to transform linking graben systems for small to moderate rift offset and low lower layer viscosity, 2) propagating but non linking and overlapping primary grabens for larger offset and intermediate lower layer viscosity, and 3) formation of multiple graben systems with inefficient rift propagation for high lower layer viscosity. The transition between the linking (Mode 1) and non-linking mode (Mode 2) is controlled by the trade-off between the rift offset, the strength of brittle-ductile coupling, and the amount of strain weakening. The mode transition from overlapping non-connecting rift segments (Mode 2) to distributed deformation (Mode 3) is mainly controlled by the viscosity of the lower layer and can be understood from minimum energy dissipation analysis arguments.

1. Introduction

[2] The way individual faults and rift segments link up is a fundamental aspect of lithosphere extension and continental break-up. There has consequently been widespread interest in the step-over regions between segments, also referred to as transfer zones [Rosendahl, 1987; Morley et al., 1990], relay structures [Larsen, 1988; Peacock and Sanderson, 1994], accommodation zones [Bosworth, 1985], or overlap zones [Childs et al., 1995], as the nature of rift interaction is determined by the way extension is accommodated in these structurally complex regions.

[3] The inherent 3D nature of the problem has mostly restricted our understanding of the underlying processes to observational evidence, semi-analytical techniques and analogue modeling studies. Most of the existing insight on rift interaction derives from studies that were directed toward understanding the origin of the orthogonal transform fault that occurs between mid-oceanic ridges [Gerya, 2012, and references therein]. Although the methodology and setup used varied significantly in the different approaches used, two end-member modes have generally been recognized: a “connecting” and an “overlapping” mode [e.g.,Choi et al., 2008]. In the connecting mode, the ridge segments are connected by a transform-like fault whereas in the overlapping mode, often called overlapping spreading center [e.g.,Hieronymus, 2004; Tentler, 2007; Acocella, 2008], the ridge segments develop a hook-shaped configuration as they overlap and bend toward each other.

[4] Little is known about what controls those specific types of interaction. Even though many of the existing analogue studies simulating offset extensional segments are limited by mechanical anisotropies or basement discontinuities [e.g., Courtillot et al., 1974; Mauduit and Dauteuil, 1996; Acocella et al., 1999] in the interaction zone that have no direct equivalent in nature, it is apparent that the geometry of interaction zones and linkage pattern of ridge segments are largely dependent on their initial configuration [Pollard and Aydin, 1984; Tentler and Acocella, 2010]. In many analogue models, however, the model behavior is dominated by tensile strain [Acocella, 2008; Tentler, 2003a, 2003b, 2007; Tentler and Acocella, 2010] whereas rifting behavior is dominated by shear deformation.

[5] Using an elastic damage rheology, Hieronymus [2004]suggests that shear damage by strain weakening is a controlling factor for contrasting styles of mid oceanic ridge spreading geometries. However, for modeling large finite strains, a frictional-plastic strain weakening rheology is more suitable.Choi et al. [2008]use 3D numerical models with an elasto-visco-plastic rheology to show that the style of ridge connectivity depends on the ratio of thermal stress to spreading-induced stress and on the rate of strain weakening. While the models developed in the mid-oceanic ridge context provide important insights on the possible controls and modes of rift interaction, the emphasis on thermal stress [Oldenburg and Brune, 1975; Choi et al., 2008] makes them less appropriate for the study of rifting in the continental lithosphere where the temperature gradient is not very large.

[6] A number of three dimensional forward models have only recently been developed [Gerya and Yuen, 2007; Braun et al., 2008; Petrunin and Sobolev, 2008; Thieulot, 2011] that can be used to study the 3D evolution of rifts and passive margins. Existing 3D numerical studies on continental rifting have focused on margin plateau formation [Dunbar and Sawyer, 1996], pull-apart basins [Katzman et al., 1995] and rift propagation [e.g., van Wijk and Blackman, 2005] but until now few 3D numerical studies have investigated rift interaction in the continental crust and lithosphere.

[7] In a recent study of 3D extension in a single layer brittle system we investigated the effect of spacing between rift segments and the amount and onset of strain weakening on the mode of rift interaction [Allken et al., 2011]. In this earlier work we demonstrated that for different combinations of these parameters, three modes of interaction are obtained: 1) grabens with a single relay zone, 2) grabens with one or more secondary step-over graben segments, and 3) large offset grabens with no significant segment interaction.

[8] These models were, however, limited by the absence of brittle-ductile coupling, which is known to provide a first order control on the structural style of rifting, where the localization of deformation is strongly linked to the viscosity of the lower layer [Buck, 1991; Huismans et al., 2005; Buiter et al., 2008]. The control of brittle-ductile coupling on deformation mode has been confirmed using analogue [Brun, 1999] and numerical experiments [Buck et al., 1999; Huismans et al., 2005]. Analytical mode transition criteria [Huismans et al., 2005; Buiter et al., 2008] based on a minimum dissipation analysis, predicted the following modes of deformation: 1) pure shear, 2) multiple conjugate or parallel shear zones, 3) two shear zones, 4) a single shear zone forming an asymmetric basin. According to these studies, the viscosity of the lower layer, and the rate and degree of plastic strain weakening control the selection of the mode of deformation.

[9] Here we explore how the initial offset between rift segments, frictional-plastic strain weakening of the upper crust and the viscosity of a ductile lower layer influence the symmetry, evolution, and structural style of rifting as well as the mode of rift interaction in a brittle-ductile coupled system. In the following we first present the numerical model, the model parameters, and the model setup. The model results are then presented documenting the effect and interaction of brittle-ductile coupling, rift offset, strain weakening parameters. Three different modes are distinguished: a connecting mode, an overlapping non-connecting mode and a distributed pure shear mode. Model behavior is subsequently analyzed and compared to the predictions of minimum dissipation analysis in the discussion.

2. Model Description

2.1. Governing Equations

[10] On geological timescales, the Earth's lithosphere deforms at a sufficiently low rate that inertial forces can be neglected (i.e. Reynolds number of the flow is zero). The momentum equation, ignoring inertial effects is given by:

display math

where σ is the stress tensor, ρ is the mass density, and g is the acceleration due to gravity. The flow is assumed to be incompressible, which implies zero divergence of the velocity field:

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The stress tensor σ can be split into a spherical part −p1 and a deviatoric part s:

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where the pressure p is given by

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For a Newtonian fluid, the deviatoric stress tensor is related to the velocity gradient through the dynamic viscosity μ as follows:

display math

where inline image is the strain rate tensor given by:

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In what follows the effects of temperature are not considered.

[11] Equations (1)(3), (5) and (6) form a closed set of equations, which allow us to compute the velocity and pressure:

display math
display math

[12] The continuity equation (8) is replaced by another equation, based on a relaxation of the incompressibility constraint which is expressed as:

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where λis the so-called penalty parameter [Hughes, 2000] that can be interpreted (and has the same dimension) as a bulk viscosity. It is equivalent to say that the material is weakly compressible. It can be shown that if λ is chosen to be a relatively large number, the continuity equation · v = 0 will be approximately satisfied in the finite element context. Equation (9) is used to eliminate the pressure in equation (7) so that the mass and momentum equation become:

display math

A new three dimensional Arbitrary Lagrangian Eulerian (ALE) fully parallel Finite Element code, FANTOM [Thieulot, 2011] is used to solve the above equations. Earth materials are highly nonlinear so that the value of the viscosity μ in equation (10) depends on the state variables v and p, and an iterative procedure is implemented to find the solution of this equation.

2.2. Rheology

[13] The rheological behavior of the upper crust is approximated with a pressure dependent Mohr-Coulomb yield criterion [Ranalli, 1995]. The strength σp is given by:

display math

where C is the cohesion, ϕ is the angle of internal friction, P is the dynamic pressure (mean stress) and θ is the Lode angle [see Thieulot, 2011, Appendix B]. Brittle failure is approximated by adapting the viscosity to limit the stress that is generated during deformation, using the viscosity rescaling method, implemented in the plastic model during the finite element matrix building process [Fullsack, 1995; Willett, 1999]. The lower layer follows a linear Newtonian viscous flow law with a constant viscosity, μv (Figure 1b).

Figure 1.

(a) Model setup showing box of dimension 210 km × 210 km × 30 km representing the brittle upper crust overlying a viscous lower crust. Extensional boundary conditions of 0.5 cm/yr are applied on 2 opposite sides of the box. In the models, two weak regions each dimensioned 2.63 km × 210 km × 1.30 km are placed at two ends of the box. In models 12–16, the offset Δ between the 2 weak seeds, is increased by a multiple of h, from 2h to 6h. (b) Rheological profile implemented: Mohr-Coulomb plasticity in the upper crust and a fixed viscosity,μv in the lower crust. (c) Frictional plastic strain weakening behavior of the upper crust. The yield stress value is constant until the strain ϵ reaches the threshold value ϵ1. For ϵ < ϵ1, the cohesion and angle of friction are constant at C0 and ϕ0. When ϵ1 < ϵ < ϵ2, the material strain weakens, i.e. C and ϕ decrease linearly from C0 and ϕ0 to Csw and ϕsw. Beyond ϵ2, cohesion and angle of friction remain constant at Csw, ϕsw.

2.3. Strain Weakening

[14] Finite strain, which is computed and stored on a three dimensional cloud of points of self-adapting density, is used to include the effects of strain weakening. The cohesionC and the angle of friction, ϕ are both functions of the accumulated strain ϵ (Figure 1c). For ϵϵ1, C and ϕ are set to C0 and ϕ0 respectively. As extension proceeds, the cloud of points accumulates strain so that when the strain in a cell reaches ϵ1, the material starts to strain weaken, i.e. the cohesion C and the angle of friction ϕ decreases linearly with strain until a final strain weakened value ϕsw is reached at ϵ = ϵ2. The yield strength in the final strain weakened state is given by:

display math

3. Model Setup

3.1. Material Layout and Boundary Conditions

[15] Our experimental set-up aims at modeling crustal extension. The model domain is a rectangular cuboid of sizeLx × Ly × Lz (Figure 1a), containing a frictional-plastic material of densityρ (the upper crust) overlying a viscous layer (the lower crust) of the same density. The upper crust is characterized by its plasticity parameters C0, Csw, ϕ0, ϕsw along with the strain softening thresholds ϵ1 and ϵ2, while the lower crust is characterized by its viscosity μv (Figure 1b). The numerical model is non-dimensional and can therefore be scaled to represent a variety of situations [Allken et al., 2011].

[16] Orthogonal extension is applied to the system, through the following boundary conditions: free slip on faces y = 0, y = Ly and at the bottom of the model domain (z = 0), imposed extensional velocities, vext, on faces x = 0 and x = Lx, and a free surface at the top of the domain. The values for model parameters can be found in Table 1.

[17] To initiate deformation, one or two weak seeds are included. Strain in the weak seeds is set to strain weakened values at the beginning of the experiments. The weak seeds, of size one quarter the length of the domain, are placed at the base of the frictional-plastic upper crust. The geometry of the weak seeds in the numerical models was purposely simple, in order to study the basic first order features of propagating rift segment interaction. The seeds may be taken to represent inherited weaknesses in the crust. Natural inheritance may exhibit more complex and unstructured characteristics that will require further investigation.

3.2. Strain-Weakening Considerations

[18] In all experiments the initial cohesion and the angle of friction of the upper crust are fixed. We choose Csw and ϕsw so that R ≥ 1 is a uniquely defined constant given by:

display math

Assuming the angle of friction to be small enough, a first-order Taylor expansion of the sine and cosine terms can be carried out inequations (11) and (12). This leads to:

display math

Note that the strain weakening ratio Ris to first-order independent of the pressure (and by extension of depth). In what follows, we use the ratioRto characterize the amount of strain-weakening which is present in the upper crust:

display math

The values of the model parameters used can be found in Table 1.

3.3. Model Nomenclature

[19] Six sets of models were run to test the sensitivity to brittle-ductile coupling, to variable rift offset, and to strain-weakening parameters (seeTable 2).

[20] 1. In the first set of models (1–3), we test the influence of brittle-ductile coupling in models with a single discontinuous seed, for lower layer viscosities,μv = 1 × 1019; 1 × 1020 and 1 × 1021 Pa.s.

[21] In all the other models, we used two discontinuous weak seeds, offset by a distance Δ, where Δ is defined as a multiple of the brittle layer thickness h (Figure 1a).

[22] 2. Models 4–11 test the effect of viscosity of the lower crust for a fixed offset Δ = 5 h on mode interaction.

[23] 3. Models 12–15 with a moderately weak lower crust test the sensitivity to varying rift offset Δ, which is systematically increased by h.

[24] 4. The next set of models tests the sensitivity to the amount of strain weakening R = 2, 3, 5, for rift offset Δ = 2 h, 3 h, 4 h, 5 h, 6 h.

[25] 5. Models 17–19 demonstrate the sensitivity to strong lower crust for variable offset Δ (2 h, 4 h, 6 h).

3.4. Numerical Considerations

[26] The computational grid is composed of 160 × 160 × 23 = 588,800 elements resulting in a mesh resolution of about 1.3 km. Trilinear velocity - constant pressure elements are used, along with a penalized formulation. The symmetric sparse matrix is solved using the massively parallel IBM sparse matrix solver WSMP [Gupta et al, 1997] on 64 cores on a Cray XT4. Typical time step size δt = 20 kyr were used, and each simulation took about 1.5 days to run.

4. Results

4.1. Influence of Brittle-Ductile Coupling

4.1.1. Models With a Single Seed

[27] We first test the effect of varying the viscosity of the lower layer of the numerical model. In models 1–3 (Figure 2), a single weak seed has been placed at the base of the brittle layer. Plastic strain weakening parameters are fixed (R = 5) and the viscosity of the lower layer is set to 1 × 1019 Pa.s, μv = 1 × 1020 Pa.s, and μv = 1 × 1021 Pa.s for models 1, 2 and 3 respectively.

Figure 2.

Models 1–3: Exploring model behavior on a simple case with a single seed (R = 5). Elevation, z (km) and cross-section of strain rate (s−1) when brittle-ductile coupling is (a) very weak (μv = 1019 Pa.s), (b) weak (μv = 1020 Pa.s) and (c) strong (μv = 1021 Pa.s). (d) Graph showing the variation of the rate of rift propagation, vy (cm/yr) with viscosity of the lower layer, μv (Pa.s), where vy is the rate at which the material, at a given point along strike, start to strain weaken (ϵ = ϵ1).

[28] In models 1–3, deformation initially localizes in two conjugate shear zones rooted in the weak seed region forming an angle of 45° with the horizontal. In model 1 at t = 1 Ma (Figure 2a) a symmetric graben has formed which has propagated efficiently through the model domain. In model 2 (Figure 2b), at t = 3 Ma, an asymmetric half graben has developed, which has propagated more than halfway across the domain. The width of the graben gradually decreases as the rift advances into the unextended domain because extension of shear zones starts later in this area. In model 3 (Figure 2c), deformation initially localizes in the weak region. Further extension favors the formation of additional grabens over the propagation of the initial rift graben along strike. At t = 10 Ma, the rift has propagated slightly and deformation is distributed over three grabens with a characteristic spacing. At the other end of the model domain, the deformation is distributed approximating pure shear, forming several small grabens. Symmetry is conserved throughout the model evolution. The degree of localization and the rate of rift propagation in these models are a direct function of the viscosity of the lower layer. Both decrease with increasing viscosity, μv (Figure 2d).

4.1.2. Variation of Viscosity for a Fixed Offset, Δ = 5 h (Models 4–11)

[29] In a previous study [Allken et al., 2011], we demonstrated that for single layer systems, the transition between modes of interaction (connecting vs non connecting grabens) occurs at a moderate offset of Δ = 5 h. For this critical offset, we now explore how the strength of the coupling to a viscous layer affects the mode of rift interaction. The plastic strain weakening parameters are fixed (R = 5), and the viscosity μv of the lower crust is varied in small increments between 1019 and 1021 Pa.s in models 4–11 (Figure 3).

Figure 3.

Models 4–11, with Δ = 5 h and R = 5, where the viscosity μv of the lower crust ranges from 1019 Pa.s to 1021 Pa.s. (left) Strain accumulated after a representative time: the red regions are the parts of the system that have reached strain weakened values. (right) Strain rate at the same time (as strain) showing the part of the system which is actively deforming at this given time. The models 4–11 are shown at different times, as the rate of propagation depends on μv.

[30] In model 4 (Figure 3a) (μv = 1019 Pa.s), the grabens propagate and connect after 2.5 Ma through an oblique transform fault forming an angle of 61° with the primary grabens. In model 5 (Figure 3b) the viscosity of the lower layer is increased to 2.5 × 1019 Pa.s. The transform fault linking the primary grabens forms at t = 3.2 Ma. At this stage the grabens have propagated further than in model 4. The transform shear zone is consequently more oblique making an angle of 71° with the original trend of the grabens. In model 6 (Figure 3c), at μv = 5.5 × 1019 Pa.s, linkage of the primary grabens is even further delayed and occurs when the tips of the propagating grabens are aligned, leading to full transform linkage shear zone perpendicular to the trend of the primary grabens. In model 7 (Figure 3d) when viscosity is increased to 6 × 1019 Pa.s, the grabens initially propagate, curve around the central region, and do not connect. Model 8 (Figure 3e) at μv = 1 × 1020 Pa.s shows similar behavior to model 7. The grabens in this case propagate even further before curving around the central region. When the viscosity is increased to 2.5 × 1020 Pa.s in model 9 the primary grabens do not seem to interact significantly and propagate largely independently (Figure 3f). In model 10 (Figure 3g), at μv = 5.0 × 1020 Pa.s, the deformation is even more distributed than in model 9 with the formation of additional shear bands adjacent to the boundary of the model. In model 11 (Figure 3h), when viscosity is very high (μv = 1 × 1021 Pa.s), the propagation of the grabens is halted. Deformation is now more distributed throughout the model domain and new grabens are formed at a characteristic distance from the original primary grabens.

[31] As viscosity of the lower layer increases, the mode of interaction between the rift segments changes from connecting, to overlapping non-connecting and finally to distributed deformation for high viscosities.

4.2. Weak Lower Crust: Sensitivity to Rift Offset and Strain-Weakening Ratio

[32] Next we test the sensitivity of the brittle-ductile coupled system to varying offset and strain weakening ratio with a constant lower layer viscosity ofμv = 1020Pa.s. Models 12–15 explore the consequence of varying rift offset for a constant strain-weakening ratio,R= 5, whereas models 12-R2, 13-R2, 14-R2, 15-R2, 16-R2 and 12-R3, 13-R3, 14-R3, 15-R3, 16-R3 test the sensitivity to the amount of strain-weakening.

4.2.1. Variation of Offset

4.2.1.1. Offset Δ = 2 h (Model 12)

[33] In model 12 (Figure 4), the two weak seeds placed at the base of the plastic layer are offset by a distance Δ = 2 h. At t = 0.5 Ma (Figure 4d), deformation initially localizes in 2 conjugate shear zones rooted in the weak seeds, forming an angle of approximately 45° with the horizontal. The innermost plastic shears with opposing dip are almost aligned at the surface. At t = 1.0 Ma (Figure 4e), the faults propagate along strike into the area without a seed, as the initial shear zones start to strain weaken leading to the formation of grabens. In the central region between the grabens deformation is initially diffuse. At t = 1.5 Ma (Figure 4f), as strain weakening sets in, strain localizes preferentially in the innermost shear zones and the outermost plastic shears are abandoned allowing for largely asymmetric deformation in both graben segments. A change of polarity between the normal faults can thus be observed in the more advanced stages of rift propagation. There is a slight flank uplift as the viscous material from the lower crust layer flows into the necking zone of the plastic layer. At t = 2.0 Ma (Figure 4g), the main active plastic shear zones connect the two asymmetric half graben segments in one continuous graben structure. The central area where the primary plastic shears with opposing dip link is characterized by distributed accumulation of strain.

Figure 4.

Evolution of model 12, where the weak seeds are offset by Δ = 2 h and the brittle-ductile coupling is weak. 3D view of deformed domain after 2 Ma of extension, showing (a) free surface elevation superimposed on the frictional-plastic upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain. (d–g) Top view of elevation, juxtaposed with side view of cross-section of strain rate after 0.5 Ma, 1 Ma, 1.5 Ma and 2 Ma.

4.2.1.2. Offset Δ = 3 h (Model 13)

[34] In model 13 (Figure 5), the two weak seeds placed at the base of the plastic layer are offset by a distance Δ = 3 h. At t = 0.6 Ma (Figure 5d), deformation initially localizes to form two shear zones rooted in the weak seeds. At t = 1 Ma (Figure 5e), two symmetric rift grabens are formed. At t = 1.6 Ma (Figure 5f), the rift grabens propagate in unextended region. At this stage, strain preferentially localizes in the innermost shear zones leading to a loss of symmetry and a change of polarity between the normal faults formed in the two primary graben segments. At 2.0 Ma (Figure 5g), the rifts curve toward one other and are almost connected in a single continuous structure. As in model 12 deformation in the linkage area is distributed.

Figure 5.

Evolution of model 13, where the weak seeds are offset by Δ = 3 h and the brittle-ductile coupling is weak. 3D view of deformed domain after 2 Ma of extension, showing (a) free surface elevation superimposed on the frictional-plastic upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain. (d–g) Top view of elevation, juxtaposed with side view of cross-section of strain rate after 0.6 Ma, 1 Ma, 1.6 Ma and 2 Ma.

4.2.1.3. Offset Δ = 4 h (Model 14)

[35] In model 14 (Figure 6), the two weak seeds placed at the base of the plastic layer are offset by a distance Δ = 4 h. At t = 1 Ma, rift grabens are formed in the weak seed regions (Figure 6d). In this case, the rifts initially propagate independently into the central area. As in the previous models the feedback effect owing to strain weakening causes strain to accumulate in the innermost shear zones, forming asymmetric grabens. At t = 3 Ma (Figure 6f), the rifts have propagated almost halfway across the domain and as deformation accumulates in the central region, the tip of the rifts segments curve around the central region. At t = 4 Ma (Figure 6g) as the central region strain weakens a dextral transfer fault links the primary grabens.

Figure 6.

Evolution of model 14, where the weak seeds are offset by Δ = 4 h and the brittle-ductile coupling is weak. 3D view of deformed domain after 4 Ma of extension, showing (a) free surface elevation superimposed on the frictional-plastic upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain. (d–g) Top view of elevation, juxtaposed with side view of cross-section of strain rate after 1 Ma, 2 Ma, 3 Ma and 4 Ma.

4.2.1.4. Offset Δ = 5 h (Model 15)

[36] In model 15 (Figure 7), the two weak seeds placed at the base of the plastic layer are offset by a distance Δ = 5 h. As in the previous models, strain accumulates on the innermost shear zones, forming asymmetric grabens. At t = 2 Ma (Figure 7e), the asymmetric rift segments start to propagate largely independently of each other into the non seed region. At t = 3 Ma (Figure 7f), the rifts propagate along each other, curving slightly around the central region. At t = 4 Ma (Figure 7g), the primary grabens propagate further, forming two hook-shaped faults around the central region which is left intact.

Figure 7.

Evolution of model 15 where the weak seeds are offset by Δ = 5 h and the brittle-ductile coupling is weak. 3D view of deformed domain after 4 Ma of extension, showing (a) free surface elevation superimposed on the frictional-plastic upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain. (d–g) Top view of elevation, juxtaposed with side view of cross-section of strain rate after 1 Ma, 2 Ma, 3 Ma and 4 Ma.

4.2.2. Effect of Strain Weakening

[37] In all previous models, a strain weakening ratio R = 5 was used. To investigate the sensitivity of the model behavior to the amount of strain weakening, models 12–16 were run with R = 2 and R = 3 (Figure 8). As described above for R= 5, the rifts propagate and connect efficiently for offsets Δ = 2 h, 3 h and 4 h, while a strike-slip dominated feature is promoted for Δ = 4 h, and rift linkage is inefficient for Δ > 4 h. Decreasing the strain weakening ratio toR = 3 only affects the style of rift linkage for intermediate offset Δ = 4 h. In this case the transform linkage structure is suppressed and more diffuse strain accumulation takes place in the linkage area. Decreasing the strain weakening ratio even further to R = 2 has a significant impact on rift linkage. The most striking difference is observed for an offset Δ = 4 h. In these conditions the rifts do not connect and at t = 3.6 Ma, start to curve around the central region, exhibiting similar behavior to model 15 (R = 5, Δ = 5 h).

Figure 8.

Influence of strain weakening on final structure when the lower crust is weak (μv = 1020 Pa.s). A different amount of strain weakening (R = 2, 3 and 5) has been applied in each row, which shows the structure formed for different offsets (Δ = 2 h–6 h). Different times are shown for the different offsets because the larger the offset, the more time it takes for the rifts to connect, if at all.

4.3. Strong Lower Crust: Variation of Offset

[38] In the last set of models 17–19, we examine the effect of a strong lower crust (μv = 1021 Pa.s) on rift mode for various rift offsets (Δ = 2 h, 4 h, 6 h) with strain weakening ratio, R = 5 (Figure 9).

Figure 9.

Model 17–19: Overview of the evolution of model behavior for different rift offset (offset Δ = 2 h, 4 h, 6 h) of the initial rift segments when coupling between the brittle and the ductile layer is very strong (μv = 1021 Pa.s).

[39] Model 17–19 show all similar behavior with little dependence on the offset of the weak seeds. Deformation at t = 4 Ma initially localizes in the weak seeds to form asymmetric primary grabens above the weak seeds. At t = 6 Ma deformation is distributed, with the coeval deformation of the primary grabens and the formation of secondary shear bands. The secondary shears form initially symmetric graben structures at a characteristic distance of about 4h from the primary grabens. Strain weakening of secondary plastic shear zones leads to asymmetry in the secondary grabens. For t ≥ 8 Ma lateral propagation of both primary and secondary graben structures leads to complex interaction structures in the central region. The detailed rift linkage structure in the central domain depends on the offset between the regularly spaced graben structures on either side of the model domain. When the offset of the initial weak seeds is a multiple of the evolving characteristic spacing between the grabens, rift propagation leads to continuous linking structures as in model 18 and 19 for t > 10 Ma. Complex linkage structure result when the primary graben offset is a fraction of the characteristic spacing (model 17, t > 10 Ma).

[40] Brittle-ductile coupling provides a strong control on the localization of deformation. Spectral analysis of the free surface elevation (Figure 10) for models 17–19 with offsets 2 h, 4 h and 6 h, indicate a characteristic spacing of about 70–75 km. For each of these three rift offsets, the dominant topographic wavelength for the evolving rift system is around these values. This indicates that the wavelength between the graben structures is independent of the position and offset of the initial weak seeds and mostly controlled by the viscosity of the lower layer.

Figure 10.

Y-averaged spectra of the (x, z) free surface elevation for models 17, 18 and 19, showing dominant wavelength at 70–75 km.

5. Discussion

[41] Our results indicate three distinct modes of interaction between two interacting rift segments for brittle-ductile coupled crustal systems in three dimensions. The modes, summarized inFigure 11, are: 1) oblique to transform linking graben systems for small to moderate rift offset and low lower layer viscosity, 2) propagating but non linking and overlapping primary grabens for larger offset and intermediate lower layer viscosity, and 3) formation of multiple graben systems with inefficient rift propagation for high lower layer viscosity.

Figure 11.

Summary of modes predicted (including some models not all shown in the paper), illustrating how the 3 modes of rift interaction are influenced by the viscosity, μv, the strain weakening ratio, R and the offset Δ and the trade-off between those parameters. The connecting mode (M1: blue), is obtained for small Δ, lowμvand is favored by low R. The overlapping non-connecting mode (M2: green) can be subdivided into a hook-shaped mode (M2a: dark green) and a propagating mode (M2b: light green) modes. Mode 2 is obtained for larger Δ and intermediateμv and is favored by low R. The distributed pure shear mode (M3: yellow) is obtained for high μv.

[42] Strain weakening, rift offset, and brittle ductile coupling provide the main controls on these modes. The degree to which the primary grabens localize deformation, propagate and link depends on: 1) the efficiency of strain accumulation and resulting strain weakening above the weak seeds, which leads to the formation of localized plastic shears, 2) the offset of the primary grabens, which affects the efficiency of strain accumulation in the linkage zone versus along strike propagation, and 3) the viscosity of the lower viscous layer that controls the efficiency of localized versus distributed strain accumulation in the brittle-ductile coupled system. The trade-off between these factors determines the relative efficiency of localized versus distributed strain accumulation.

[43] In all models, strain initially accumulates above the weak seed, leading to the formation of the primary grabens. Distributed deformation occurs in the region ahead of the primary grabens. Rift propagation is achieved when the accumulated strain in this area reaches strain weakening values. At the same time, distributed strain accumulates in the rift linkage area and in the areas offset from the primary grabens. Whether rift linkage (mode 1), rift propagation (mode 2), or off axis distributed deformation is preferred depends on the relative efficiency of strain accumulation and resulting strain weakening in each of these areas.

5.1. Factors Controlling Mode of Rift Interaction

5.1.1. Effect of Brittle-Ductile Coupling

[44] The first two sets of models (Figures 2 and 3) demonstrate that viscosity of the lower layer plays a key role in the evolution and mode of deformation of extensional grabens and the nature of their interaction with other graben segments.

[45] At low viscosities when the brittle-ductile coupling is very weak, strain localizes efficiently in the shear zones above the weak seed. As viscosity of the lower layer (and strength of coupling) increases, deformation becomes increasingly distributed throughout the system, demonstrating an inverse relationship between strain accumulation and strength of brittle-ductile coupling. The weaker the coupling, the faster the strain weakening threshold is reached in the initial shear zones. This leads to fast and efficient rift propagation. This is apparent in Models 1–3, where in the time that is required for a rift to propagate throughout a very weakly coupled system (e.g. Model 1), only a fraction of that distance is covered by rifts with a more strongly coupled system (e.g., Model 2 and 3) (Figure 2). The different rates of rift propagation explain why the variation in the width of the rift along strike becomes larger as μvincreases. At any given time, the shear zones formed above the weak seed have accommodated more extension and have been advected sidewards by extension, than the shear zones formed subsequently by rift propagation in the previously homogeneous regions. This lateral variation in the initiation and formation of the rift results in the formation of a V-shaped rift, which becomes more pronounced for higher viscosities.

[46] The influence of the viscosity of the lower layer on rift segment interaction is studied with models 4–11 (Figure 3). Varying the viscosity of the lower layer, μv, in small increments allows three contrasting modes of rift interaction. At low viscosities, each of the rift segments localizes strain and propagates efficiently. Distributed strain accumulating in the rift linkage area results in efficient connectivity between the rift segments. At these low viscosities all the extension is accommodated within the innermost shear zone, which is favored as the rifts exert an influence over each other leading to asymmetry.

[47] As viscosity increases, the rift becomes more symmetric as the outermost shear zone also accumulate strain and strain weaken. Beyond a certain viscosity, strain accumulation in the linkage area is less efficient and no longer sufficient to cause the rifts to link. The competition between strain accumulation in the central region and along strike causes the rifts to curve around the central region.

[48] At high viscosities, deformation is distributed and the rifts no longer interact with each other. Localized grabens form with a characteristic wavelength of about 5 times the brittle layer thickness. While the distributed mode is only obtained for high viscosities, the selection of a connecting mode or an overlapping non-connecting mode also depends on offset between rift segments and the amount of strain weakening.

5.1.2. Offset Δ Between Rift Segments

[49] When the offset, Δ, is small, the linkage area coincides with the propagation area facilitating efficient connectivity. When the offset is large, distributed strain accumulates in each of these areas leading to competing domains of strain localization.

[50] Offset between the rift segments is the second major control on the mode of rift interaction. When the offset Δ, is small, the zones of distributed deformation in front of the propagating rift segments are close enough to overlap. When the brittle-ductile coupling is weak, strain accumulates more efficiently in the rift linkage area than along strike promoting linkage. Beyond an offset of around Δ = 4 h (depending onμvand R), the strain accumulated in the central region is no longer sufficient to induce linkage and an overlapping hook-shaped mode of rift interaction is obtained.

5.1.3. Amount of Strain Weakening R

[51] For intermediate offsets, a decrease in the amount of strain weakening causes a change in the mode of rift interaction from transform linking mode (M1) to overlapping non linking mode (M2), as a result of the decrease in the efficiency of strain weakening in the linkage area. Using a different approach in which weakening was achieved through energy-dependent damage,Hieronymus [2004] showed similarly that weakening favors a shift in mode interaction from transform fault to overlapping spreading center.

[52] For low strain weakening ratios (R = 3–4) the transition from linking to non-linking graben systems occurs at viscosities in the range [1–5] × 1019 Pas, whereas for higher strain weakening ratios (R = 5) this mode transition occurs at viscosities of [0.5–1] × 1020Pas. This can be explained by the trade-off between strain weakening in the linkage area and the resistance to localization in this area provided by brittle ductile coupling.

5.2. Interplay Between Mode Transition Controls

[53] Figure 11illustrates the primary controls on rift interaction and their trade-off effects. The mode of interaction between rift segments in the system depends on the combined effects of brittle-ductile coupling, the onset and magnitude of strain-weakening, and the offset between the rift segments. While increasing the offset, Δ, tends to cause a shift from linking mode 1 to non-linking mode 2, decreasing strain weakening has the same effect for intermediate offsets (Figure 8). Increasing viscosity on the other hand, for a given offset and strain weakening ratio R, can cause the system to switch from localized mode (M1) to distributed mode (M2 and M3) of interaction (Figure 3). Oblique to transform linking mode (M1) is favored by small offset, a large amount of strain weakening, and low viscosity. Localized but non-linking grabens mode (M2) is obtained for intermediate to large offsets and intermediate viscosities, and is favored by a small amount of strain weakening. The distributed deformation mode (M3) results when the brittle-ductile coupling with the lower layer is strong.

[54] Other factors may also affect the selection of these modes. Strain rate or velocity of rifting provide a similar control on brittle-ductile coupling as lower layer viscosity [e.g.,Huismans et al., 2005; Buiter et al., 2008; Choi et al., 2008], with increased rate of extension leading to distributed deformation modes. Second, the onset of strain weakening has been demonstrated to affect the efficiency of rift linkage [e.g., Allken et al., 2011]. This suggests that different strain weakening mechanisms characterized by contrasting onset, magnitude, and rate of strain weakening may lead to varying efficiency of rift linkage. Last, the existence of the ductile lower layer also impacts on the modes of interaction. Comparison of the models presented here with our earlier work on 3D extensional systems that included a plastic layer on a fixed base [Allken et al., 2011] only demonstrates this. The inclusion of a fluid viscous layer allows for isostatic compensation, resulting in shallower compensated basins and subdued rift flank topography. Furthermore brittle-ductile coupling accounts for the new features observed in the models presented here, i.e. the orthogonal transform fault and the distributed mode of rifting.

5.2.1. Emergence of Orthogonal Transform Faults

[55] Orthogonal transform faults only occur when coupling is weak, for intermediate offsets (4 h or 5 h) at the transition between mode 1 and 2. This feature is observed when the conditions are such that the zones of distributed deformation from each rift segment overlap and reach strain weakening values when the rifts have each propagated halfway throughout the model. Orthogonal transform faults only occur within a narrow parameter space, for a given combination of rate of rift propagation, largely controlled by the viscosity of the lower layer and strain weakening parameters. Using a purely elastic damage rheology, Hieronymus [2004] suggests that the formation of transform faults depends on the rate of shear damage formation relative to that of ridge propagation. This is consistent with the results presented here which indicate that orthogonal transform faults can form by gradual focusing of diffuse damage.

5.2.2. Distributed Pure Shear Mode

[56] The mode transition between overlapping non-connecting rift segments to distributed deformation can be understood using minimum energy dissipation arguments. Mode transition can be predicted if it is assumed that the preferred mode minimizes the internal rate of energy dissipation. Earlier analytical work [Huismans et al., 2005; Buiter et al., 2008] predicts the transition viscosity image beyond which the distributed mode is favored over the strain weakened symmetric graben mode. Adapting this formulation (see Appendix A) to the rheology used in our models gives us the following transition viscosity image

display math

For reasonable parameter values (Appendix A) the mode transition viscosity can be calculated for varying amount of strain weakening ratio R (Table 3). The values of the predicted transition viscosity image are consistent with the model results presented here that indicate the mode transition for viscosities ranging from μv = 2.5 × 1020 to 5 × 1020 Pa.s (Figures 3 and 11).

Table 1. Model Parameters
SymbolMeaningValue
LxLength of model along x-axis210 km
LyLength of model along y-axis210 km
LzLength of model along z-axis30 km
hpThickness of plastic layer15 km
hvThickness of viscous layer15 km
ρDensity2800 kg.m−3
gAcceleration due to gravity9.81 m.s−2
C0Initial cohesion20 MPa
ϕ0Initial angle of friction15°
ϵ1Strain softening threshold0.25
ϵ2Onset of full strain weakened state1.25
vextExtensional velocity0.5 cm/yr
λPenalty factor1032 Pa.s
Table 2. Model Parameters Used: Seed Arrangement, Viscosity of Lower Layer, μv, and Strain Weakening, R
FigureModel NumberSeed Arrangementμv (Pa.s)R
Figure 21Single seed1.0 × 10195
Figure 22Single seed1.0 × 10205
Figure 23Single seed1.0 × 10215
Figure 34Δ = 5 h1.0 × 10195
Figure 35Δ = 5 h2.5 × 10195
Figure 36Δ = 5 h1.0 × 10195
Figure 37Δ = 5 h6.0 × 10195
Figure 38Δ = 5 h1.0 × 10205
Figure 39Δ = 5 h2.5 × 10205
Figure 310Δ = 5 h5.0 × 10205
Figure 311Δ = 5 h1.0 × 10215
Figure 412Δ = 2 h1.0 × 10205
Figure 513Δ = 3 h1.0 × 10205
Figure 614Δ = 4 h1.0 × 10205
Figure 715Δ = 5 h1.0 × 10205
Figure 816Δ = 6 h1.0 × 10205
Figure 812-R2Δ = 2 h1.0 × 10202
Figure 812-R3Δ = 2 h1.0 × 10203
Figure 813-R2Δ = 3 h1.0 × 10202
Figure 813-R3Δ = 3 h1.0 × 10203
Figure 814-R2Δ = 4 h1.0 × 10202
Figure 814-R3Δ = 4 h1.0 × 10203
Figure 815-R2Δ = 5 h1.0 × 10202
Figure 815-R3Δ = 5 h1.0 × 10203
Figure 816-R2Δ = 6 h1.0 × 10202
Figure 816-R3Δ = 6 h1.0 × 10203
Figure 917Δ = 2 h1.0 × 10215
Figure 918Δ = 4 h1.0 × 10215
Figure 919Δ = 6 h1.0 × 10215
Table 3. Transition Viscosity image as a Function of Strain Weakening Ratio R
Rimage (Pa.s)
22.67 × 1020
33.46 × 1020
43.79 × 1020
54.11 × 1020

5.3. Model Limitations

[57] 1. Temperature dependent rheologies, although available as an option, have not been included. Transient variations in the depth of the brittle-ductile are expected to affect rift evolution. Given that no conductive cooling and associated deepening of the brittle-ductile transition is included, the models presented here should be interpreted in the context of moderate to fast extended crustal systems at the advective limit.

[58] 2. The boundary conditions applied in these models, orthogonal extension, are kept simple. Natural systems may extend by combinations of both oblique extension and varying amounts of extension along strike. We do not expect that varying amounts of extension along strike would significantly change the modes observed in our simple models. This, however, needs further investigation. Oblique extension with respect to inherited heterogeneity (weak seeds) should result in oblique rift modes but is beyond the scope of this paper.

[59] 3. Surface processes (erosion and deposition) are not included. These are expected to have an effect on the structural style of sedimentary basin formation but are beyond the scope of the present paper.

6. Conclusions

[60] We use state-of-the-art large deformation 3D numerical models to study the structural style of crustal extension. We examine the controls on the style and geometry of rift linkage between rift segments during extension of crustal brittle-ductile coupled systems, and test the sensitivity of varying the viscosity of the lower layer, the offset between the rift basins, and the amount of strain-weakening on the efficiency of rift linkage and rift propagation, and the style of extension. We conclude that:

[61] 1. The major controls on the mode of rift interaction in brittle-ductile coupled crustal systems are, in decreasing order of importance: 1) strength of brittle-ductile coupling, 2) rift offset, and 3) the amount of strain weakening.

[62] 2. For small to moderate rift offset and low lower layer viscosity, Mode 1, oblique to transform linking graben is preferred. For larger offset and intermediate lower layer viscosity, Mode 2 with propagating but non linking and overlapping primary grabens is promoted, and for high lower layer viscosity, Mode 3 with the formation of multiple graben systems and inefficient rift propagation is obtained.

[63] 3. The mode transition between the linking Mode 1 and non-linking Mode 2 is controlled by the trade-off between the rift offset, the strength of brittle-ductile coupling and the amount of strain weakening.

[64] 4. The transition from Mode 2 overlapping non-connecting rift segments to Mode 3 distributed deformation is mainly controlled by the viscosity of the lower layer and can be understood using minimum energy dissipation arguments.

Appendix A:: Transition Between Modes Predicted From Dissipation Analysis

[65] Dissipation analysis in 2D [Huismans et al., 2005] predicted that as viscosity increases, the system will deform in asymmetric graben (AP) mode, symmetric graben (SP) mode and pure shear (PS) mode respectively, the criteria for mode selection being based on the minimization of energy dissipation. At a transition between two modes, the total dissipation in these modes are equal:

display math

where inline imageis half the rate of internal viscous-plastic dissipation per unit volume and inline image is the total rate of work against gravity. Modes 1, 2, and 3 represent AP, SP and PS modes. It was considered in this study that mode selection could be estimated with sufficient accuracy from the internal dissipation inline image alone. The internal dissipation is the sum of the viscous dissipation inline image and the plastic dissipation inline image, which depends on the strength of the frictional-plastic layer,σ. Since the strength of the plastic layer is pressure-dependent, we take the mean strength,σm:

display math

At transition between modes:

display math

Internal dissipation for:

Pure shear mode:

display math

Symmetric graben mode:

display math

where hb is the boundary layer and v is the extensional velocity.

[66] At the transition between the symmetric graben mode and the pure shear mode:

display math
display math
display math
display math

[67] Let a be inline image

display math

Given that the Lode angle θ lies between − π/6 < θ < π/6,

For ϕ = 15 and 2 < ϕsw < 10,

display math
display math

Replacing equation (A12) in equation (A9) gives a transition viscosity image

display math

Acknowledgments

[68] This work was funded through Norwegian Research Council Grant 177489/V30 to Huismans, who also acknowledges support through an EU International Reintegration Grant and from the Bergen Center for Computational Science. We thank Suzon Jammes and Philippe Steer for valuable discussions and constructive comments that helped significantly improve the manuscript. Taras Gerya and an anonymous reviewer are thanked for their quick reviews and helpful suggestions.

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